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Tame dynamics and robust transitivity chain-recurrence classes versus homoclinic classes


Authors: C. Bonatti, S. Crovisier, N. Gourmelon and R. Potrie
Journal: Trans. Amer. Math. Soc. 366 (2014), 4849-4871
MSC (2010): Primary 37C20, 37D30, 37C29, 37G25
DOI: https://doi.org/10.1090/S0002-9947-2014-06261-2
Published electronically: May 8, 2014
MathSciNet review: 3217702
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Abstract: One main task of smooth dynamical systems consists in finding a good decomposition into elementary pieces of the dynamics. This paper contributes to the study of chain-recurrence classes. It is known that $ C^1$-generically, each chain-recurrence class containing a periodic orbit is equal to the homoclinic class of this orbit. Our result implies that in general this property is fragile.

We build a $ C^1$-open set $ \mathcal {U}$ of tame diffeomorphisms (their dynamics only splits into finitely many chain-recurrence classes) such that for any diffeomorphism in a $ C^\infty $-dense subset of $ \mathcal {U}$, one of the chain-recurrence classes is not transitive (and has an isolated point). Moreover, these dynamics are obtained among partially hyperbolic systems with one-dimensional center.


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  • [A] Flavio Abdenur, Generic robustness of spectral decompositions, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 2, 213-224 (English, with English and French summaries). MR 1980311 (2004b:37032), https://doi.org/10.1016/S0012-9593(03)00008-9
  • [AS] R. Abraham and S. Smale, Nongenericity of $ \Omega $-stability, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 5-8. MR 0271986 (42 #6867)
  • [B] C. Bonatti, Survey: Towards a global view of dynamical systems, for the $ C^1$-topology, Ergodic Theory Dynam. Systems 31 (2011), no. 4, 959-993. MR 2818683 (2012j:37029), https://doi.org/10.1017/S0143385710000891
  • [BC] Christian Bonatti and Sylvain Crovisier, Récurrence et généricité, Invent. Math. 158 (2004), no. 1, 33-104 (French, with English and French summaries). MR 2090361 (2007b:37036), https://doi.org/10.1007/s00222-004-0368-1
  • [BD$_1$] Christian Bonatti and Lorenzo J. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math. (2) 143 (1996), no. 2, 357-396. MR 1381990 (97d:58122), https://doi.org/10.2307/2118647
  • [BD$_2$] Christian Bonatti and Lorenzo Díaz, On maximal transitive sets of generic diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. 96 (2002), 171-197 (2003). MR 1985032 (2007d:37017), https://doi.org/10.1007/s10240-003-0008-0
  • [BD$_3$] Christian Bonatti and Lorenzo Díaz, Robust heterodimensional cycles and $ C^1$-generic dynamics, J. Inst. Math. Jussieu 7 (2008), no. 3, 469-525. MR 2427422 (2009f:37020), https://doi.org/10.1017/S1474748008000030
  • [BD$_4$] Christian Bonatti and Lorenzo J. Díaz, Abundance of $ C^1$-robust homoclinic tangencies, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5111-5148. MR 2931324, https://doi.org/10.1090/S0002-9947-2012-05445-6
  • [BD$_5$] Ch. Bonatti and L. J. Díaz, Fragile cycles, J. Differential Equations 252 (2012), no. 7, 4176-4199. MR 2879727, https://doi.org/10.1016/j.jde.2011.12.002
  • [BDP] C. Bonatti, L. J. Díaz, and E. R. Pujals, A $ C^1$-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2) 158 (2003), no. 2, 355-418 (English, with English and French summaries). MR 2018925 (2007k:37032), https://doi.org/10.4007/annals.2003.158.355
  • [BDV] Christian Bonatti, Lorenzo J. Díaz, and Marcelo Viana, Dynamics beyond uniform hyperbolicity, Encyclopaedia of Mathematical Sciences, vol. 102, Springer-Verlag, Berlin, 2005. A global geometric and probabilistic perspective; Mathematical Physics, III. MR 2105774 (2005g:37001)
  • [BV] Christian Bonatti and Marcelo Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math. 115 (2000), 157-193. MR 1749677 (2001j:37063a), https://doi.org/10.1007/BF02810585
  • [Ca] Maria Carvalho, Sinaĭ-Ruelle-Bowen measures for $ N$-dimensional [$ N$ dimensions] derived from Anosov diffeomorphisms, Ergodic Theory Dynam. Systems 13 (1993), no. 1, 21-44. MR 1213077 (94h:58102), https://doi.org/10.1017/S0143385700007185
  • [CP] S. Crovisier and E. Pujals, Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms, ArXiv:1011.3836.
  • [Co] Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133 (80c:58009)
  • [F] John M. Franks, Homology and dynamical systems, CBMS Regional Conference Series in Mathematics, vol. 49, Published for the Conference Board of the Mathematical Sciences, Washington, D.C., 1982. MR 669378 (84f:58067)
  • [G] Nikolaz Gourmelon, Generation of homoclinic tangencies by $ C^1$-perturbations, Discrete Contin. Dyn. Syst. 26 (2010), no. 1, 1-42. MR 2552776 (2010m:37033), https://doi.org/10.3934/dcds.2010.26.1
  • [KH] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374 (96c:58055)
  • [M] Ricardo Mañé, Contributions to the stability conjecture, Topology 17 (1978), no. 4, 383-396. MR 516217 (84b:58061), https://doi.org/10.1016/0040-9383(78)90005-8
  • [PS] Enrique R. Pujals and Martín Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math. (2) 151 (2000), no. 3, 961-1023. MR 1779562 (2001m:37057), https://doi.org/10.2307/121127
  • [S] M. Shub, Topological transitive diffeomorphism on $ T^4$, Lect. Notes in Math. 206 (1971), 39-40.
  • [N] Sheldon E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 101-151. MR 556584 (82e:58067)
  • [W] Lan Wen, Homoclinic tangencies and dominated splittings, Nonlinearity 15 (2002), no. 5, 1445-1469. MR 1925423 (2003f:37055), https://doi.org/10.1088/0951-7715/15/5/306

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Additional Information

C. Bonatti
Affiliation: CNRS - IMB, UMR 5584, Université de Bourgogne, 21004 Dijon, France
Email: bonatti@u-bourgogne.fr

S. Crovisier
Affiliation: CNRS - LMO, UMR 8628. Université Paris-Sud 11, 91405 Orsay, France
Email: sylvain.crovisier@math.u-psud.fr

N. Gourmelon
Affiliation: IMB, UMR 5251, Université Bordeaux 1, 33405 Talence, France
Email: Nicolas.Gourmelon@math.u-bordeaux1.fr

R. Potrie
Affiliation: CMAT, Facultad de Ciencias, Universidad de la República, Uruguay
Email: rpotrie@cmat.edu.uy

DOI: https://doi.org/10.1090/S0002-9947-2014-06261-2
Received by editor(s): January 10, 2012
Received by editor(s) in revised form: December 10, 2012
Published electronically: May 8, 2014
Additional Notes: The authors were partially supported by the ANR project DynNonHyp BLAN08-2 313375.
Article copyright: © Copyright 2014 American Mathematical Society

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